Abstract
Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the KoksmaHlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain KoksmaHlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual KoksmaHlawka theorem. We show that the space of functions of bounded Dvariation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.
Original language  English 

Pages (fromto)  773797 
Number of pages  25 
Journal  Journal of Complexity 
Volume  31 
Issue number  6 
Early online date  16 Jun 2015 
DOIs  
Publication status  Published  01 Dec 2015 
Keywords
 HardyKrause variation
 Harman variation
 Integral geometry
 KoksmaHlawka theorem
ASJC Scopus subject areas
 Algebra and Number Theory
 Statistics and Probability
 Numerical Analysis
 Control and Optimization
 Applied Mathematics
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Florian Pausinger
Person: Academic